THE CHIRAL PHASE TRANSITION IN QCD:

CRITICAL PHENOMENA

AND LONG WAVELENGTH PION OSCILLATIONS

KRISHNA RAJAGOPAL

Lyman Laboratory of Physics, Harvard University,

Cambridge, MA, USA. 02138

E-mail:

ABSTRACT

In QCD with two flavours of massless quarks, the chiral phase transition is plausibly in the same universality class as the classical four component Heisenberg magnet. Therefore, renormalization group techniques developed in the study of phase transitions can be applied to calculate the critical exponents which characterize the scaling behaviour of universal quantities near the critical point. As a consequence of this observation, a quantitative description of the universal physics of the chiral phase transition in circumstances where the plasma is close to thermal equilibrium as it passes through the critical temperature can be obtained. This approach to the QCD phase transition has implications both for lattice gauge theory and for heavy ion collisions. Future lattice simulations with longer correlation lengths will provide quantitative measurements of the various exponents and the equation of state for the order parameter as a function of temperature and quark masses. Present lattice simulations already allow many qualitative tests. In a heavy ion collision, the consequence of a long correlation length would be large fluctuations in the number ratio of neutral to charged pions. Unfortunately, we will see that because of the explicit chiral symmetry breaking introduced by the masses of the up and down quarks, no equilibrium correlation length gets long enough for this phenomenon to occur. In the third chapter of this article, I discuss attempts to model the dynamics of the chiral order parameter in a far from equilibrium QCD phase transition by considering quenching in the O(4) linear sigma model. I argue, and present numerical evidence, that in the period immediately following the quench long wavelength modes of the pion field are amplified. This could have dramatic phenomenological consequences in heavy ion collisions. I review various recent theoretical advances — attempts to include expansion and relax the quench assumption; attempts to include quantum mechanical effects. Long wavelength pion oscillations have now been seen in a number of simulations in which different assumptions are made. However, all of the theoretical treatments involve idealizations and assumptions, and should only be regarded as qualitative guides to what can happen. It is up to experimentalists to determine whether or not such phenomena occur; detection in a heavy ion collision would imply an out of equilibrium chiral transition.

To appear in Quark-Gluon Plasma 2, edited by R. Hwa, World Scientific, 1995.

April, 1995. HUTP-95-A013

## 1 Introduction

Shortly after the discovery that QCD is asymptotically free and that therefore quarks are almost noninteracting at short distances, Collins and Perry noted that this means that QCD at high temperature or high density is qualitatively different than at low temperature and density. At high temperatures () or high baryon number density () QCD describes a world of weakly interacting quarks and gluons very different from the hadronic world in which we live. This raises the possibility of phase transitions as the temperature or density is increased. While the subject of phase transitions as a function of increasing density has interesting applications in the physics of neutron stars and heavy ion collisions, in this article I will limit myself to discussing the physics of a QCD plasma with zero baryon number density. There are at least two qualitative differences between such a plasma at and at . First, as Collins and Perry discussed, at low temperatures one has a plasma of quarks and gluons confined in mesons and baryons, while at high temperatures the quarks and gluons are deconfined. Second, at low temperatures a condensate spontaneously breaks chiral symmetry while at high temperatures the interactions among the quarks and anti-quarks are weak, no such condensate exists, and chiral symmetry is manifest. Thus, in studying the phase structure of QCD as a function of increasing temperature, there are two possible phase transitions to consider.

Studying the QCD phase transition(s) has a certain intrinsic appeal. The question involved is simple enough that it can be stated in a sentence: What happens if you take a chunk of any substance around you and heat it up until it is one hundred thousand times hotter than the centre of the sun? While the question is simple, if one tries to answer it experimentally by colliding heavy ions at ultra-relativistic energies, the phenomena seem at first glance complicated and intractable. And yet, using techniques from different areas of theoretical physics — quantum field theory, particle physics, nuclear physics, lattice gauge theory, condensed matter physics, and cosmology — much progress is possible. Quantitative predictions for processes in which a plasma cools through the QCD phase transition in thermal equilibrium can be extracted, and striking phenomena with qualitative signatures may occur if the plasma is far from equilibrium as it cools through the transition.

In addition to having intrinsic appeal, the QCD phase transition is of interest from several different points of view. First, there can be no doubt that it occurred throughout the universe about a microsecond after the big bang. Second, it is reasonable to hope that in a heavy ion collision of sufficiently high energy, a small region of the high temperature phase is created which then cools through the phase transition. Third, lattice gauge theory is well suited to calculating the equilibrium properties of QCD at high temperatures. From all these perspectives, it is important to learn as much as can be learned analytically about the phase transition, relying as much as possible only on fundamental symmetries and universality arguments and as little as possible on specific assumptions and models. In Chapter 2, following original work done with Wilczek, we will see that it is possible to use this approach to learn much about the equilibrium QCD phase transition. In studying heavy ion collisions, however, it is worth considering the possibility that the transition process may be far from thermal equilibrium. Without thermal equilibrium, it is much harder to be quantitative both because the phenomena are more varied and because there is no notion of universality. We will nevertheless find that the distinctive qualitative signature of the QCD phase transition in heavy ion collisions discussed in Chapter 3 — the amplification of long wavelength oscillations of the pion field — can only occur if the plasma is far from thermal equilibrium as it experiences the QCD phase transition.

The physics of QCD at high temperatures is an enormous subject with many facets, which I will not attempt to review here. The reader interested in entering the voluminous literature might consider starting with the review by Gross, Pisarski, and Yaffe, the review by McLerran, the book by Shuryak, the previous collection of articles edited by Hwa, recent reviews by Müller, recent conference proceedings in the Quark Matter series, or other articles in this volume.

The remainder of this introduction has two goals. First, a brief discussion of the confinement/deconfinement phase transition seems in order, since the bulk of this article is concerned with the chiral phase transition. Second, I will give a very brief introduction to some of the physics relevant for a discussion of the QCD phase transition in heavy ion collisons, in order to set the stage for the third chapter.

Let us recall the arguments which lead to the conclusion that the confinement/deconfinement phase transition is first order in pure gauge theory and is smoothed into a crossover when light quarks are present. In studying gauge theory in thermal equilibrium at a temperature , the partition function is given by a Euclidian path integral

(1) |

over a slab of spacetime which is infinite in the three spatial directions and is thick in the direction. where the are representation matrices for the fundamental representation of . The path integral is over gauge field configurations which satisfy periodic boundary conditions . The Euclidian action is given by

(2) |

The expectation value of operators is given by

(3) |

We will be particularly interested in the Wilson loop operator

(4) |

for a loop at fixed which loops around the periodic time direction once. denotes path ordering. This operator, conventionally called the Polyakov loop, is of interest because the free energy of a single static quark is

(5) |

where is the free energy of the vacuum. If , inserting a quark into the ensemble costs an infinite free energy. This implies that the flux cannot be screened — the state of the fields is changed all the way out to spatial infinity and costs an infinite free energy. Therefore, indicates that the theory is in the confining phase. If , then is finite and the theory is in a deconfined phase characterized by Debye screening.

Under a gauge transformation

(6) |

transforms as

(7) |

so is invariant when is periodic. Local observables are invariant under those aperiodic gauge transformations

(8) |

where is an element of the centre of the gauge group. This is not true of the Polyakov loop, which transforms under (8) according to

(9) |

Therefore, is an order parameter. Going from to requires the spontaneous breaking of a symmetry — in this case, spontaneous breaking of . Hence, the realization of the symmetry directly determines whether or not the gauge theory is in a confining phase.

The confinement/deconfinement phase transition as a function of temperature in gauge theory is a transition in which a global symmetry is spontaneously broken. If the order parameter varies continuously (but non-analytically) from for to for , then the transition is second order. If, on the other hand, is discontinuous at , the transition is first order. Following Landau, we write an effective Lagrangian for at long length scales whose terms are determined by the symmetry of the order parameter

(10) |

where the effective potential is

(11) |

In general, and are arbitrary functions. To look for a second order phase transition, we expand them about the origin giving

(12) |

where , , and are functions of temperature and we have specialized to . Taking , we see that if were zero, the minimum of the potential would increase continuously from to as went through zero. However, with the term present will always jump discontinuously away from zero as is lowered. In fact, no cases are known in which there is a second order phase transition involving an order parameter whose symmetries allow a cubic invariant in the free energy. The cubic term which is present because the symmetry is forces the phase transition to be first order.

The QCD phase transition is an inherently non-perturbative phenomenon. Perhaps the best long term strategy for studying it in full (non-universal) detail from first principles is to use lattice gauge theory. Following Wilson, the theory can be formulated on a lattice and the Euclidian path integral (1) can be evaluated using Monte Carlo techniques. This involves evaluating expectation values of operators by importance sampling — one chooses configurations weighted by a known probability measure — and is therefore limited to studying equilibrium situations in which no time evolution occurs. The prediction of Svetitsky and Yaffe described above that the pure phase transition is first order has been confirmed in lattice gauge theory simulations.

Including dynamical quarks in the partition function changes the situation entirely. is nonzero even in the confined phase. This can be understood heuristically by noting that a static quark source can always be screened by virtual pairs in the ensemble. It can also be understood more formally by showing that after integrating out the fermions, the effective action for the gauge fields includes a term which breaks the symmetry. No order parameter is known for the confinement/deconfinement transition in gauge theory when dynamical fermions are present. This would seem to preclude the existence of a second order transition. A first order transition is logically possible, and presumably is obtained for heavy quarks. For two species of light quarks, on the other hand, lattice gauge theory simulations show no signs of a first order deconfinement transition. increases smoothly as a function of increasing temperature and no discontinuities are apparent. As in the ionization of a gas, there is no sharply defined transition, but rather a smooth change from one regime to another.

Arguing only from fundamental symmetries, we have found that the phase transition is first order in pure gluon QCD. We found an order parameter, but its symmetry precluded a second order transition. In the presence of dynamical quarks, we were unable even to find an order parameter. Had we found a second order transition, we could do much more. Powerful renormalization group techniques could be brought to bear on the problem, superceding the mean field analysis above. When there are massless quarks present, while there is no order parameter for the confinement/deconfinement transition, there is an order parameter for the chiral phase transition. In Chapter 2, following arguments first advanced by Pisarski and Wilczek which are similar in philosophy to those above, we will discover that for two species of massless quarks the chiral phase transition can be second order. This will have many implications and quantitative consequences.

All the results of Chapter 2 describe quantities observable in thermal equilibrium where there is no time evolution. As we have discussed, this makes them ideal for comparison with lattice “experiments”. Working on the lattice, one has more freedom to adjust the parameters of the theory than in the real world. “Experimentalists” can choose what quark masses to use, while experimentalists cannot. We will see that this is unfortunate, because in order to observe the critical phenomena discussed in Chapter 2 on the lattice or in the real world, it will be desirable to use smaller up and down quark masses than nature does. This is possible on the lattice, although difficult in practice in current simulations. The future for “experimentalists” is bright.

What about experimentalists? Experimentalists hope to study the physics of the QCD phase transition in relativistic heavy ion collisions. We will argue in Chapter 2 that if the plasma in these events stays in equilibrium, it is unlikely that the chiral transition will have a dramatic signature. Of course, it is still possible to hope to detect the difference between and . It is just that the physics at will be smooth. In studying heavy ion collisions, however, we really should be open to the possibility that the plasma cools through the transition without staying in thermal equilibrium. This will be our concern in Chapter 3. We will find that it is possible that long wavelength oscillations of the pion field are excited, leading to the production of clusters of pions within which the number ratio of neutral pions is far from 1/3. This is a potentially dramatic signature of the chiral phase transition in relativistic heavy ion collisions in which thermal equilibrium is not maintained.

As previously mentioned, the second goal of this introduction is to sketch some standard ideas about what may happen in ultra-relativistic heavy ion collisions. We will not review this enormous field comprehensively. References to reviews wherein the interested reader can find both discussions of greater depth and further references will be given. The reader should also consult other articles in this volume.

Let us recall a few qualitative facts which are known empirically from studying high energy proton-proton collisions with multiple production of secondary particles, and which are presumed to apply to high enough energy collisions between nuclei also. The momentum distribution of the secondary particles which could in general be a function of the longitudinal and transverse momenta of the particles (,) and the centre of mass energy of the incident protons () is in fact observed to be independent of for larger than about and the invariant cross-section factorizes according to

(13) |

, , and are the energy, momentum and mass of a particular secondary particle. The Feynman variable is given by

(14) |

The transverse momentum distribution falls rapidly with increasing and the mean transverse momentum is . (Events in which hard parton-parton scatterings give rise to jets with large occur about once per million collisions. Most high energy proton-proton collisions yield secondary particles with small transverse momenta.) To discuss the longitudinal momentum distribution it is useful to define a variable , called rapidity, where

(15) |

One reason why is convenient is that boosting in the beam direction corresponds simply to adding a constant to . Another reason is that for ,

(16) |

where the pseudorapidity is easily measured since it depends only on the polar angle . A third useful property of is that

(17) |

We will work in the centre of mass frame, where corresponds to and corresponds to where

(18) |

Feynman postulated that at large there is a central rapidity region (, or, equivalently, ) where is a constant. In this region,

(19) |

and, integrating over and using (17),

(20) |

Thus, there is a central rapidity region in which the multiplicity per unit rapidity is approximately constant. This result has indeed been seen in proton-(anti)proton collisions. The qualitative features found in high energy hadron-hadron collisions — low and a central rapidity plateau — are expected to characterize ultra-relativistic nucleus-nucleus collisions also.

In a relativistic heavy ion collision of sufficiently high energy, the decay products divide naturally into two regions. In the centre of mass reference frame, the fragments of the incident nuclei are found at high (positive and negative) rapidity, while the central rapidity region is characterized by approximately constant multiplicity per unit rapidity and by low baryon number density. At early times, this central rapidity region experiences boost invariant longitudinal expansion in which the momentum and position of particles are related in such a way that the rapidity is given by

(21) |

where is the time and is the position coordinate in the beam direction. Mean physical quantities depend only on the proper time

(22) |

and transverse position, and are independent of .

Describing the fragmentation regions requires consideration of QCD at nonzero temperature and baryon number density, and is not discussed in this article. We limit ourselves to addressing the physics of the “hot vacuum” in the central rapidity region. How high an energy is sufficient for there to be a reasonably clean separation between distinct fragmentation and central rapidity regions? The parameter of interest is , the extent in rapidity of one of the fragmentation regions. is not well known — estimates range from 4 to 8. Even if , present fixed target heavy ion collision experiments at the AGS and SPS do not access sufficient rapidities for there to be a baryon poor central rapidity region. At the AGS, and at the SPS . In the coming generation of colliding beam heavy ion accelerators, the situation will be much improved. At the Relativistic Heavy Ion Collider currently under construction at Brookhaven National Laboratory, it is expected that Au-Au collisions with with multiplicities of at least 1400 per unit rapidity will be achieved. When the LHC is built at CERN and is run as a Pb-Pb collider, collisions with and with multiplicities of at least 2600 per unit rapidity will be possible. These multiplicity estimates are qualitative — they are based on extrapolating from data from proton-nucleus collisions, and results from event generators suggest that they could be too low. At RHIC () the central rapidity region will have essentially zero baryon number density if and at the LHC () there will be such a region for . Even if there are baryons in the central region at RHIC, the number ratio of mesons to baryons is estimated to be . (At the LHC, this ratio is expected to be .) Thus, analyzing the physics of QCD at finite temperature and zero baryon number density will be of relevance in describing the physics of the central rapidity regions of heavy ion collisions at RHIC and the LHC.

What happens in the central rapidity region? The standard description involves three phases — a pre-equilibrium phase, a hydrodynamic evolution phase during which thermal equilibrium is assumed, and a hadronization phase. There have been a number of different approaches to treating the earliest phase of a heavy ion collision. For example, in the string model, one imagines that the incident nuclei pass through each other and colour flux tubes, or strings, are drawn out between the receding fragmented nuclei. These strings then break apart by quark–anti-quark pair production, and realizations of the scenario are based on Monte Carlo simulation of this process. This scenario has been reviewed by Białas and Czyź. It runs into difficulties at energies high enough that the string density becomes large. At high enough energies, it should be possible to describe the earliest phase of the evolution in terms of parton interactions by perturbative QCD. This approach, called the parton cascade model, has begun to yield results. One starts with distribution functions for the partons in the nucleons of the incident nuclei, and follows the quark-gluon interactions in the framework of renormalization group improved perturbative QCD by solving transport equations for the parton distributions in phase-space with Monte Carlo methods. The results of this work give support at the parton level to the idea that a thermalized state is formed. Quarks thermalize more slowly than gluons because the relevant QCD cross-sections are larger by a factor of two or three. Also, for both quarks and gluons, a thermal distribution of momenta is achieved more quickly than chemical equilibrium (equilibrium number density) is achieved. (It is not clear whether there will be enough time in a RHIC collision for quark chemical equilibrium to be established.) It is of interest that at the latest times considered by Geiger (), the baryon number is in fragmentation regions of high rapidity, and the central rapidity region contains partons with thermalized momenta which are expanding longitudinally with little transverse motion. While these calculations are far from the end of the story of the pre-equilibration physics, they do provide justification for supposing that in an energetic enough collision (perhaps at RHIC) there is a low baryon number central rapidity region in which particle momenta are in local thermal equilibrium at .

If equilibrium is achieved, and while it is maintained, the evolution of the plasma can be described hydrodynamically. The matter is treated as a relativistic fluid described by smoothly varying four-velocity, pressure, temperature, and energy density without reference to the microscopic description in terms of partons. This approach has a long history, but perhaps the first quantitative treatment was that of Bjorken. The review by Blaizot and Ollitrault is a useful introduction to the subject. Bjorken used the observation that the central rapidity region is approximately boost invariant to find solutions to the hydrodynamic equations describing the longitudinal expansion and consequent cooling of the plasma. Blaizot and Ollitrault review various numerical simulations of the hydrodynamic equations which include transverse expansion.

In using the hydrodynamical approach one assumes that thermal equilibrium is valid throughout the transition and until the hadrons no longer interact. “This is an assumption which is hard to justify in any satisfying way.” It is particularly difficult to see how long wavelength degrees of freedom can stay in equilibrium as the plasma expands and cools through the phase transition. Given the uncertainties, in considering the physics of the chiral phase transition in relativistic heavy ion collisions we will discuss both the phenomena that would occur if the plasma stays in equilibrium and those that could characterize a far from equilibrium transition.

In the final phase of a heavy ion collision, regardless of what has happened earlier, hadrons, photons, and leptons impinge upon an experimentalist’s detector. How is the experimentalist supposed to unravel what has gone on before? The zeroth order question is how can she determine whether for a brief instant a small region of quark-gluon plasma was created? Many signatures have been proposed for the production of a region of quark-gluon plasma in a heavy ion collision and its consequent passage through the QCD phase transition. All have problems to overcome, arising either from background, or from insufficient theoretical understanding, or from the possibility that the signature can be mimicked by hadronic processes. This suggests that the QCD phase transition will be detected by a process where various pieces of evidence become more and more convincing rather than by the observation of a “smoking gun” signature. If the speculations of Chapter 3 are correct, however, the situation could be improved. We will see that if the chiral phase transition occurs by a process that is far enough from thermal equilibrium that it can be idealized as a quench, then coherent long wavelength oscillations of the pion field may emerge. These would manifest themselves as clusters of pions with similar rapidity in which the number ratio of neutral pions is dramatically different from 1/3. If seen, this phenomenon would be a distinctive, qualitative signature that a far from equilibrium chiral transition had occurred.

## 2 Critical Phenomena at a Second Order QCD Phase Transition

The QCD phase transition has been studied from many different perspectives using many different models and approximations. Here, however, we will attempt to learn as much as can be learned analytically about the phase transition, relying as much as possible only on fundamental symmetries and universality arguments and as little as possible on specific assumptions and models. This philosophy yields little fruit when applied to the confinement/deconfinement transition. We saw in the previous chapter that in QCD with light quarks, this transition is neither first order nor second order — it is a smooth crossover like the ionization of a gas. The chiral phase transition is another story. Wilczek has emphasized that in the chiral limit where there are two species of quarks with zero current algebra mass, the order parameter for the chiral phase transition has the same symmetry as the magnetization of a four component Heisenberg magnet, which has a second order phase transition. At an equilibrium second order phase transition, thermodynamic quantities have non-analytic behaviour determined by the physics of the modes which develop long correlation lengths at the critical point. Thus, by studying these universal characteristics in a much simpler model — the Heisenberg magnet — it is possible to learn much about the chiral phase transition in QCD.

In this chapter, which to a large extent follows work done with Wilczek, we explore the consequences of this approach to the QCD phase transition in thermal equilibrium. In the following section, we use renormalization group arguments to establish a dictionary between QCD and the magnetic system. Using this dictionary, we apply results for the static critical exponents obtained in the magnet model to QCD. In section 2.2, we discuss the behaviour of the pion and sigma masses at the transition. In section 2.3, we review recent work in lattice gauge theory which has begun to test these ideas. In section 2.4, we consider how the strange quark affects the phase transition, and again compare with lattice results. In the concluding section of this chapter, we discuss the implications of all this for cosmology and lattice gauge theory, and for heavy ion collisions under the assumption that the chiral order parameter remains close to thermal equilibrium through the phase transition.

### 2.1 QCD and the Magnet

The physics of the QCD phase transition is qualitatively different in the cases of zero, one, two, or three or more flavours of quarks. In this section we consider QCD with two species of quarks. Pisarski and Wilczek used an analysis similar to the one which follows to show that for three or more flavours of massless quarks, the chiral phase transition is first order. In section 2.4 we will discuss the effects of a third quark species, massless or massive, and for now, therefore, we postpone our discussion of the effects of the strange quark. If there are two flavours of massless quarks, the QCD lagrangian is symmetric under global chiral transformations in the group of independent special unitary transformations of the left and right handed quark fields, and a vector transformation which corresponds to baryon number symmetry. (The axial which would make the symmetry group into is a symmetry of the classical theory, but not of the quantum theory.) This chiral symmetry breaks spontaneously down to at low temperatures, and is restored at sufficiently high temperatures. The order parameter for this phase transition is the expectation value of the quark bilinear

(23) |

which breaks the symmetry when it acquires a nonzero value below some critical temperature .

At a second order phase transition, the system is at an infrared fixed point of the renormalization group. Renormalization group transformations correspond to coarsening one’s view of the world. Thus, at an infrared fixed point, physics is scale invariant. The order parameter fluctuates on all length scales and, in particular, on arbitrarily long length scales. The correlation length associated with these scale invariant fluctuations is necessarily infinite, and the correlation function which describes them is a power law. Our goal is not to describe all of the physics occurring at the transition. Rather, we limit our attention to the nonanalyticities of thermodynamic quantities. The partition function is the sum of analytic terms, and so in a system with finitely many degrees of freedom must be an analytic function of the temperature. Nonanalyticity can only arise in the infinite volume limit, and so by limiting the scope of our treatment in this way we concentrate on the physics of the long wavelength degrees of freedom. At an infrared fixed point, renormalization (i.e. coarsening one’s view) does not change the physics of the long wavelength fluctuations of the order parameter, which is therefore independent of all short wavelength physics. The nonanalytic critical phenomena are universal: they only depend on the modes with long wavelength fluctuations (and these are determined by the symmetries of the order parameter) and on the scaling behaviour near the fixed point of low dimension operators constructed from these modes.

In order to describe a second-order transition quantitatively, we must find a tractable model in the same universality class. For the chiral order parameter (23) the relevant symmetries are independent unitary transformations of the left- and right-handed quark fields, under which

(24) |

These transformations generate an symmetry, which is not quite what is needed, since it includes the axial baryon number symmetry which is not present in QCD. This problem is solved by restricting to multiples of unitary matrices with positive determinant, instead of general complex matrices. Matrices in this restricted class remain in this restricted class under the transformation (24) only if and have equal phases. Hence, the axial has indeed been removed. The matrices can conveniently be parametrized using four real parameters and the Pauli matrices as

(25) |

In fact the order parameter can be written as a four-component vector

(26) |

and the transformations (24) are simply rotations in internal space. Hence, the order parameter appropriate for the chiral phase transition in QCD with two flavours of massless quarks has the symmetries of the standard invariant Heisenberg magnet. For smaller number of components, this sort of model is a much-studied model for the critical behaviour of magnets, with the order parameter representing the magnetization of a ferromagnet or the staggered magnetization of an antiferromagnet.

In order to discuss the universal critical phenomena, it is sufficient to retain those degrees of freedom which develop large correlation lengths at the critical point. These are just long wavelength fluctuations of the order parameter, which is small in magnitude near the critical point and therefore fluctuates at little cost in energy. Thus, the most plausible starting point for analyzing the critical behaviour of a second-order chiral phase transition in QCD is the Landau-Ginzburg free energy

(27) |

Here is the temperature dependent
renormalized mass squared.
is the temperature at which vanishes.
is positive above the critical temperature and negative below it.
The quartic coupling depends smoothly on the temperature.
In this section we consider the case in which is positive;
we will discuss what happens if is not positive at
in section 2.4.
We neglect higher dimension operators, because they are irrelevant
at an infrared fixed point --- their coefficients go to zero upon
renormalization toward the infrared.^{a}^{a}a is not
irrelevant in 3-dimensions, but as long as is positive
at , can be neglected relative to
since is small near the transition.
The
term will be important in section 2.4. The
symmetry breaking pattern we want is (equivalently,
)
below the
transition which is indeed what we find at the minimum of the
potential for positive . This model has been studied in depth
for arbitrary and spatial dimension , and the existence of
an infrared stable fixed point of the renormalization group has been
established. Hence, it is a model for a second order QCD chiral
phase transition for two massless quarks.

Before proceeding, let us briefly outline the the physics of the theory given by (27) in thermal equilibrium above, below, and at the critical temperature at which the second order phase transition occurs. At temperatures above , the direction in internal space in which is oriented is correlated only over spatial distances of order a finite correlation length; over longer distances all orientations of are equally probable. Thus, is disordered and . At temperatures below , is ordered: is nonzero and selects a particular direction (defined to be the direction) in internal space. Whereas the theory has an explicit symmetry, the equilibrium configuration is not symmetric — the symmetry is spontaneously broken. Fluctuations of about in the and directions are described by differing correlation functions. The explicit symmetry implies that could choose any direction in internal space. Below , therefore, fluctuations of in the directions (which correspond to fluctuations of the orientation of ) are massless Goldstone modes. At , , the correlation length is infinite, and the fluctuations of are scale invariant.

We have established that there is an infrared fixed point of the renormalization group in the same universality class as QCD with two massless quarks. The theories in this universality class live in an infinite dimensional parameter space spanned by , , and the coefficients of an infinite number of irrelevant higher dimension operators. The second order fixed point has some basin of attraction in this space of theories. For the rest of this chapter, we will explore the hypothesis that QCD is indeed in the basin of attraction of the infrared fixed point. We will calculate much about the QCD phase transition based on this assumption. We will also be show that it is consistent with many phenomena found to date in lattice gauge theory simulations. We can never be sure, however, that upon renormalization toward the infrared QCD is in fact driven to this fixed point. It is always a logical possibility that, say, QCD may live in a region of theory space from which infrared renormalization drives the theory into a region of negative , making the phase transition first order. Hence, the strongest statement that it is possible to make is that it is plausible that in QCD with two species of massless quarks the chiral phase transition is second order. Our strategy, then, is to establish quantitative consequences of this hypothesis and test them against lattice simulations of QCD.

When the free energy (27) is written in terms of and it looks much like the original model of Gell-Mann and Lévy with two changes: there are no nucleon fields and only three (spatial) dimensions. These two changes reflect an important distinction. We are only proposing (27) as appropriate near the second order phase transition point. This is because it is only there that we can appeal to universality — the long-wavelength behaviour of the and fields is determined by the infrared fixed point of the renormalization group, and microscopic considerations are irrelevant to it. In Euclidean field theory at finite temperature, the integral over of zero temperature field theory is replaced by a sum over Matsubara frequencies given by for bosons and for fermions with an integer. Hence, one is left with a Euclidean theory in three spatial dimensions with massless fields from the terms in the boson sums and massive fields from the rest of the boson sums and the fermion sums. Hence, to discuss the massless modes of interest at the critical point, (27) is sufficient and we do not need to introduce nucleon fields or constituent quark fields. In arriving at an effective three dimensional theory of the long wavelength fluctuations of the order parameter near , all other bosonic degrees of freedom and all fermionic degrees of freedom have been integrated out.

We have motivated a very definite hypothesis for the nature of the phase transition for QCD with two species of massless quarks, namely that it is in the universality class of the Heisenberg magnet. This means that under renormalization toward the infrared, QCD with two massless quarks is driven to the infrared fixed point of (27) and that therefore the nonanalytic behaviour of thermodynamic quantities near is the same in the two theories. This hypothesis has numerous consequences which are the subject of the rest of this chapter. In the remainder of this section, we review the resulting predictions for the critical exponents.

First, we define the reduced temperature . The exponents , , , , and describe the singular behaviour of the theory with strictly zero quark masses as . For the specific heat one finds

(28) |

The behaviour of the order parameter defines .

(29) |

and describe the behaviour of the correlation length where

(30) |

is independent of , but may depend on . The correlation length diverges as , and the correlation length exponent is defined by

(31) |

Above , where the correlation lengths are equal in the sigma and pion channels, the susceptibility exponent is defined by

(32) |

We will discuss the behaviour of the susceptibility below the transition in the following section. At the critical point, the correlation length is infinite and the correlation function is a power law. The exponent is defined through the behaviour of the Fourier transform of the correlation function at :

(33) |

The last exponent, , is related to the behaviour of the system in a small magnetic field which explicitly breaks the symmetry. Let us first show that in a QCD context, is proportional to a common quark mass . This common mass term may be represented by a matrix given by times the identity matrix. We are now allowed to construct the free energy from invariants involving both and . The lowest dimension term linear in is just , which in magnet language is simply the coupling of the magnetization to an external magnetic field . In the presence of an external field, the order parameter is not zero at . In fact,

(34) |

Thus when the symmetry is explicitly broken by a small external magnetic field or equivalently by small quark masses, the phase transition is a smooth crossover. It is, however, a very special crossover about which much can be said, because as one approaches the second order fixed point.

The six critical exponents defined above are related by four scaling relations. These are

(35) |

We therefore need values for and for the four component magnet in . These were obtained in the remarkable work of Baker, Meiron and Nickel, who carried the perturbative expansion in for the theory (27) to seven-loop order, and used information about the behaviour of asymptotically large orders, and conformal mapping and Padé approximant techniques to obtain

(36) |

Using (35), the remaining exponents are

(37) |

Since is negative there is a cusp in the specific heat at , rather than a divergence. There are other ways in which these exponents could be calculated. In , mean field theory suffices and one has and . We are interested in , however. A standard approach is to work in dimensions, evaluate the exponents order by order in , and then set . The -expansion has not been pushed to high enough order to compete in accuracy with the work of Baker, Meiron, and Nickel. Another approach is to do a finite temperature lattice simulation of the theory (27) and measure the critical exponents. This has recently been done to high accuracy by Kanaya and Kaya, who find and , in agreement with the results of Ref. [?].

To reiterate, these exponents and the other critical phenomena we discuss in subsequent sections are universal. Whether the QCD phase transition is studied experimentally, or on the lattice, or by instanton liquid techniques, or by using the Nambu–Jona-Lasinio model, or by using the (four dimensional) linear or nonlinear sigma models, or by any other means, if the hypothesis that the chiral transition is a second order transition in the universality class is correct the exponents should turn out to have the values we have described. Of course, using any technique except appealing to the simplest model in the same universality class as we have done would make it much harder to study the theory near the critical point, and hence much harder to obtain the critical exponents.

### 2.2 The Equation of State and the Pion and Sigma Masses

The expressions which define , and are actually special cases of a more general relationship between the magnetization and the magnetic field called the critical equation of state. The equation of state has been calculated to order by Brézin, Wallace and Wilson. In this section, we review the use of the equation of state to determine the behaviour of the pion and sigma masses near the critical point.

First, we must define what we mean by the “mass” of the pion and sigma. We could choose either to define the mass as an inverse correlation length or as an inverse susceptibility. We choose the latter, which is conventional in the condensed matter literature. Specifically, we define

(38) |

and

(39) |

where and . Defining “masses” as inverse correlation lengths would give different scaling behaviours for masses as functions of and . (For , however, both choices would yield the same scaling behaviour, and in the theory of interest is small.) We shall see that with the conventional choices (38) and (39), the masses can be extracted conveniently from the equation of state. It is worth noting that the masses we have defined are related only to the behaviour of spatial correlation functions in the static (equilibrium) theory. They carry no dynamical information, and should not be confused with, say, pole masses in a 4-dimensional theory. Also, we will only be able to make universal statements about how the masses scale at the transition. Normalizing the magnitudes of the masses will require using some specific model, and hence will not be universal.

The equation of state gives the magnetization as a function of and . For the rest of this chapter, we will write the order parameter as , for magnetization, keeping in mind that . Also, recall that , the external field, is proportional to the quark mass . In order to define the equation of state, we first define a shifted field . Then the equation of state is simply the relation

(40) |

This relation has been expanded to order by Brézin, Wallace and Wilson. The result can be expressed conveniently in terms of the variables

(41) |

as

(42) |

The function is universal and was calculated to order by Brézin, Wallace, and Wilson. Their result is also quoted in Ref. [3]. The units in which and are measured are chosen so that corresponds to and corresponds to the coexistence curve. Knowing , we can calculate the value of the order parameter for a given and using (42). The behaviour of the order parameter is illustrated in Fig. 1. This figure and the other ones in this section should be viewed as illustrations of qualitative behaviour rather than quantitative predictions because they are based on setting in the expression for . The values for the critical exponents themselves which we quoted in the previous section are quantitative predictions, complete with error estimates, because they are based on the much more elaborate analysis of Baker, Meiron and Nickel.

From the equation of state, we can deduce the behaviour of and at nonzero (but small) and . The masses are given by

(43) |

and

(44) |

The first relation follows directly from the definition (38), and the second follows from (39) and from assuming that , so that a small change gives a small change with . Using the equation of state, we can rewrite (43) and (44) as

(45) |

and

(46) |

Hence, can be determined by measuring the ratio at . In general, from we can find the pion and sigma masses for any and .

There are two interesting limits which we will consider explicitly. First, for and which corresponds to , we should find the full symmetry, and hence should find that the pion and sigma masses are identical. For , the function behaves as . The constant is given to in Ref. [3]. Applying (45) and (46) , we find that

(47) |

consistent with the symmetry.

We can also consider the limiting case of approaching the coexistence curve. This means taking and , which implies . In this limit, tends to a nonzero constant, and so from (44) , we obtain , a familiar result for Goldstone bosons. The behaviour of the pion mass is illustrated in Fig. 2.

The result (44) may look peculiar to a particle physicist who is more familiar with the zero temperature result

(48) |

Before considering the sigma mass, we therefore pause here to explain how (48) and (44) are related. We have seen that and that the order parameter . At zero temperature, is defined in terms of the axial current by the relation

(49) |

In the zero temperature linear sigma model, the axial current is given by

(50) |

which means that defined in (48) is simply

(51) |

This result suggests that we make the identification , which does indeed make (44) and (48) equivalent.

However, it is important to remember that using the linear sigma model at zero temperature cannot be justified by a universality argument in the way that using it near can. Hence the argument of this paragraph is not a derivation of (44) from the zero temperature result (48) . (44) is valid near while (48) is valid at . Also, in (48) is a mass in a dimensional Lorentz invariant theory, while in (44) is an inverse susceptibility in a 3 dimensional theory. We have simply shown that a reader familiar with one expression should not be surprised by the other.

The behaviour of the sigma mass at the coexistence curve is trickier to obtain than that of the pion mass. First, we note that in mean field theory () the equation of state is simply , and is easily evaluated using (46) . For at fixed the result is

(52) |

Hence, in mean field theory decreases with to a nonzero value at . However, for when fluctuations are important, the result is quite different. In words, fluctuations of the massless pions produce new infrared singularities in the longitudinal susceptibility, or, equivalently, make the sigma massless. Now, let us see how this result can be obtained from the equation of state. In the limit , while , we will see, tends to zero more slowly. Hence, the second term in (46) is dominant and gives

(53) |

The difficulty is that for , contains divergent terms like , and . These terms do not exponentiate to . Wallace and Zia in fact find the result

(54) |

Both the terms on the right side of (54) must be kept because they differ in their exponents only by order . Also for this reason, the constants and calculated by Wallace and Zia and given in Ref. [?] are only known to order even though is known to order .

Qualitatively, as is lowered at fixed , at first the term dominates and appears to be decreasing toward a nonzero value at as in the mean field result. Then, the term takes over and one finds that in fact the sigma mass goes to zero like . Recently, Anishetty et al. have done an extensive analysis of the divergence of the sigma susceptibility produced by the massless pions for and . In agreement with the -expansion result, they find that in , . The behaviour of the sigma meson mass is illustrated in Fig. 3 and Fig. 4.

In future lattice simulations, as is lowered toward zero, this behaviour should be observed. This result is an example of the power of the renormalization group techniques in obtaining universal results. If we had chosen a specific microscopic model, say that of Gocksch, or the Nambu–Jona-Lasinio model of Hatsuda and Kunihiro, we would have been able to calculate non-universal quantities far from , but would basically have been limited to using mean field theory, as those authors do. Then, we would have reached the incorrect conclusion that in the chiral limit below . Here, by restricting ourselves to calculating universal quantities, we are limited to the region near the critical point, but our results are model independent and include the effect of fluctuations.

### 2.3 Comparison with Lattice Simulations

Finite temperature lattice QCD simulations are ideally suited to testing many of the predictions made in this chapter. For a pedagogical introduction to and review of finite temperature lattice QCD simulations, see DeTar’s article in this volume. The static correlation functions of the three dimensional theory are natural objects to consider in finite temperature (Euclidean) simulations. Also, it is much easier to vary parameters like the temperature and the bare quark mass in a lattice simulation than in a real experiment. Hence, it should be possible to measure the static critical exponents of section 2.1, and the equation of state and the scaling behaviour of the pion and sigma masses of section 2.2 on the lattice.

Present simulations provide evidence that in two flavour QCD the phase transition is second order, and that the order parameter is indeed . For example, in the results of Bernard et al. shown in Fig. 5, there are no signs of any discontinuities in the expectation value of the order parameter as a function of temperature. decreases smoothly as a function of increasing temperature, as Fig. 1 leads us to expect for simulations done with nonzero quark masses if the transition is indeed second order for . All these authors have looked unsuccessfully for signals of two phases coexisting at one temperature, which would be characteristic of a first order transition. These simulations are consistent with the transition being second order, but do not rule out the possibility that in better simulations with smaller quark masses a small discontinuity could be found, indicating that the transition is in fact weakly first order. Using the zero temperature mass to set the energy scale, Bernard et al. find that the temperature at which the decrease in the chiral order parameter occurs is about , significantly lower than in simulations without dynamical fermions. Over the same range of temperatures at which the chiral order parameter decreases, the Polyakov loop expectation value increases and the specific heat is large. Also, the calculations slow down — simulations must be run for a long time to get reliable results because large fluctuations occur. This is as would be expected near a second order critical point. Thus, in QCD with 2 dynamical quarks, it seems that the crossover associated with deconfinement, for which there is no order parameter, occurs at the same temperature as the chiral transition. The chiral transition is also a smooth crossover, but we expect it to approach a second order transition as is lowered further. All of this is in marked contrast to results of lattice gauge theory simulations with three or four species of light quarks, in which there is a first order chiral phase transition with latent heat, hysteresis, and abrupt changes in the values of observables. (Our results on the order of the chiral transition — second for two massless quarks and first for three or more — are also in agreement with results obtained by treating the QCD vacuum as an instanton liquid and making a mean-field approximation.)

Fig. 6. shows the behaviour of the screening mass (inverse correlation length) for the and in the same simulation as Fig. 5. If the exponent were zero, then and (which, recall, are susceptibilities) would have the same scaling behaviour as the screening masses of Fig. 6. Since is in fact small, the screening masses can be viewed as crude approximations to and . As we expect, the sigma screening mass decreases and the pion screening mass increases as the temperature is increased, and above the two masses each increase and appear to be becoming degenerate. This is evidence that the order parameter is indeed .

So far, we have discussed various qualitative lattice results, that are in agreement with our expectations. Before turning to recent progress on more quantitative tests, it is worth enumerating the logical possibilities which could lead to a failure of the hypothesis that a second order chiral phase transition with exponents will be seen in lattice simulations. Perhaps the most likely alternative is that the transition could in fact be first order. As we will see in the next section, this occurs if the strange quark is too light. More generally, we only know that an infrared fixed point with appropriate symmetry exists. We do not know that QCD is in fact in the basin of attraction of this fixed point. A second logical possibility is that the Ginzburg region may be too small to see the true critical behaviour. Fluctuations of the order parameter are important only close to — for where , like , is not universal. Outside the Ginzburg region, (i.e. for ,) mean field theory is valid. Thus, even if the chiral phase transition does have exponents, these exponents can only be measured by simulations done for temperatures satisfying . Simulations with larger would see the mean field exponents and . , like , cannot be predicted by arguments based on universality, and must be measured on the lattice. Can be very small? This is a logical possibility. For example, in ordinary, low temperature, BCS superconductors, where is the Fermi energy. In these materials, and is minuscule. Recent numerical simulations by Kocic and Kogut of the dimensional Gross-Neveu model in which mean field exponents are seen can be interpreted as a sign that is small in this theory. In QCD, there is no analogue of the small parameter , and so there is no reason to expect to be particularly small. Nevertheless, it remains a logical possibility. To exclude this scenario, it is therefore crucial to measure the critical exponents in QCD simulations with two flavours accurately enough to demonstrate that they deviate from their mean field values. The third logical possibility is that somewhere in theory space there is another infrared fixed point to which QCD is driven under renormalization. Were this the case, the chiral phase transition would be second order but would not have exponents. Of the three logical possibilities, the third would be the most surprising.

Recently, a number of authors have begun to attempt quantitative comparisons between lattice simulations and expectations based on the hypothesis that the chiral transition is second order with exponents. DeTar has pointed out that it is possible to check that the order parameter , the temperature , and the symmetry breaking term are in fact related by a single equation of state (42). Following Binder, DeTar writes (42) in a different, but equivalent, fashion. Define

(55) |

Then, the equation of state (42) is equivalent to

(56) |

where is a universal function related to of by

(57) |

Fig. 7 is DeTar’s plot of versus for a number of simulations with various quark masses, temperatures, and lattice sizes. If (56) is correct, all the points should lie on a single curve. In producing the figure, DeTar has used the values for and , and has adjusted to to obtain the best agreement, although he notes that works comparably well. DeTar’s plot is encouraging, but not yet quantitative. For one thing, the preliminary data on the largest lattices with 12 lattice sites in the direction are not in good agreement with the other data, suggesting that finite size effects may be important. In the future, with better simulations, it will be possible to apply this analysis more quantitatively — the simulation results will be fit to (56) yielding best fit values for , and , along with error estimates.

Karsch has pioneered another way of measuring the critical exponents in a lattice simulation. From Fig. 3. we see that for , has a minimum at , which Karsch calls the pseudocritical temperature. Because , , and are related by (42) or equivalently (56),

(58) |

Karsch therefore notes that in lattice simulations, the pseudocritical coupling at which the susceptibility peaks should depend on according to

(59) |

where is related to for the second order transition with massless quarks, and neither nor are universal. Karsch and Laermann’s plot of the pseudocritical coupling in various simulations is shown in Fig. 8. Fitting the three parameters , , and yields , which is slightly higher than the value and which is in agreement with the mean field value . However, fitting and with fixed to its value yields just as good a fit. Simple inspection of the figure leads to the conclusion that it is too early to extract the value of the exponent convincingly — simulations with lighter quarks are needed.

Karsch and Laermann have evaluated both the and the susceptibilities. From these, they extract a direct measurement of the exponent . From (45) and (46), we see that the quantity which Karsch and Laermann call is given by

(60) |

For , for and for , for . Most important, however, is that for any quark mass, at — i.e. at . Curves of as a function of for different quark masses should cross at , and at that coupling they should have . Karsch and Laermann’s results are shown in Fig. 9. They find

(61) |

This is in agreement with the value and disagrees with the mean field value . To make this result even more compelling, it would be good to have curves for lighter quarks to see that they all cross at the same and , and to see decreasing toward zero as for .

One critical exponent which has not been mentioned in this section is , which is associated with the specific heat. Because , there should be a cusp in the specific heat. However, the specific heat can also have a large analytic contribution. In fact, as the temperature is raised through the chiral transition, deconfinement also occurs and the number of degrees of freedom in thermal equilibrium increases substantially, leading to a large specific heat. This large contribution to the specific heat is an analytic function of even for , while the contribution from the chiral order parameter is not. Nevertheless, because the smooth part of the specific heat is so large, extracting the non-analytic part and thus will be difficult.

By this point it should be clear that quantitative tests of the hypothesis that the QCD phase transition is in the same universality class as the magnet are just beginning to be possible. To date, lattice simulations of two flavour QCD are in qualitative agreement with this hypothesis. As lattice simulations improve, these tests will become more and more quantitative. It should soon be possible to exclude conclusively the possibility that the exponents take on mean field values. Indeed, the results of Karsch and Laermann for are close to doing this already. This would demonstrate that the simulations are close enough in temperature to that the fluctuations of the order parameter are important — i.e. the simulations are within the Ginzburg region. This would be reassuring, as it would imply that as simulations improve further the true critical behaviour will be explored, and the hypothesis that the transition is in the universality class will be tested quantitatively.

Now, some caveats. Present lattices are small. Ideally, one wants the correlation length to be large compared to the lattice spacing and small compared to the lattice size, and this requires much larger lattices. (Berera has suggested, however, that this problem can be turned into a virtue — finite size effects and finite lattice spacing effects affect the critical behaviour and therefore can be used to measure critical exponents like .) However, the fundamental reason why present simulations do not offer more quantitative tests of the predictions in this chapter is that to date quark masses have been so large that correlation lengths do not get very long at . For example, in the work of Bernard et al., the correlation length in the pion channel at is only about 2.5 lattice lengths. To really probe the critical phenomena, lighter quarks are needed. Unfortunately, longer correlation lengths are necessarily accompanied by numerical critical slowing, and this makes simulations challenging.

Recent work by the Columbia group offers another indication that lighter quark masses are necessary in order to quantitatively probe the critical phenomena. They vary the “valence quark mass” (the quark mass in the operator inserted in order to measure the order parameter) and the “sea quark mass” (the mass of the dynamical quarks in the fermion determinant) independently. cannot be made too small because to do so would slow the simulation down; can be made arbitrarily small. The first result they find is that the order parameter goes to zero as a temperature dependent power of for fixed . (All masses are in lattice units.) They interpret the power law behaviour they see for between about and as a sign of some sort of critical behaviour, but no interpretation in terms of the exponents we have discussed is possible, since these exponents are only relevant for the physical case where . They also find indications that for fixed , variations of which are accessible with their present simulations (i.e. down to ) seem to renormalize and have no other effect. (It is known that the effect of heavy quarks is to shift .) If these preliminary results are confirmed, they suggest that must be reduced further than has been possible to date in order to see the critical phenomena characteristic of discussed in this article.

Another hurdle to be overcome before lattice simulations can measure the critical properties of the QCD phase transition is that any lattice implementation of fermion fields only exhibits the full chiral symmetry in the continuum limit. Thermodynamic simulations with Wilson fermions are difficult because Wilson fermions break chiral symmetry completely. Larger lattices or an improved action are required to avoid lattice artifacts. If staggered fermions are used, one starts with four flavours of fermions which in the continuum have an chiral symmetry. On the lattice, a subgroup is all that remains. In order to study two flavours of fermions, one takes the square root of the fermion determinant in the lattice action. It is not at all clear what this does to the lattice chiral symmetries. What is known is that all three pions become light only in the continuum limit. Finite lattice spacing effects leave one pion light, while making two of them heavy. A transition in which a symmetry breaks to would be in the universality class of the magnet, as Boyd et al.